Variational approach to finding the cost-optimal trajectory

There are different approaches to define the path which is optimal in the sense of a construction cost. Such problems on practice are usually solved by various heuristic procedures. To get a theoretically justified result, one can derive an integral cost functional under certain assumptions and use variational principles. Thus, the classical problem of the calculus of variations is obtained. The necessary condition for the minimum of such a functional has the form of the integro-differential equation.
This paper describes a numerical algorithm for solving this equation, which is based on the prominent and detally studied in the literature shooting method. Under additional assumptions via Schauder fixed point principle the existense of the solution is proved. The problem of the uniqueness of the solution is studied. A numerical example is provided.